Quick Answer: Tolerance Statistic Result

The verified regression model predicted G3 final grade from 13 predictors: G1, G2, studytime, failures, absences, age, Medu, Fedu, traveltime, famrel, freetime, goout and health. The overall model was strong, with R = .923, R² = .852, adjusted R² = .849, F(13, 635) = 281.518, and p < .001. The main Tolerance Statistic result is that all predictors stayed above the common warning threshold of tolerance = .20. The lowest tolerance values were for G1 = .238 and G2 = .238. These values are low compared with other predictors, but they are still acceptable under the tolerance ≥ .20 rule.

Hypothesis-style decision: The null assumption says the regression model does not have harmful multicollinearity, meaning H₀: Tolerance Statistic ≥ .20 for all predictors. The alternative assumption says at least one predictor has problematic multicollinearity, meaning H₁: Tolerance Statistic < .20 for at least one predictor. Since the lowest tolerance is .238, the tolerance-based decision is to retain the null assumption of acceptable multicollinearity. However, G1 and G2 should still be discussed carefully because they have the lowest tolerance and the highest VIF cross-check values.

What Is Tolerance Statistic?

Tolerance Statistic is a regression diagnostic used to detect multicollinearity among independent variables. It tells how much of a predictor’s variance is not explained by the other predictors in the model. If the Tolerance Statistic is high, the predictor has a large amount of unique information. If the Tolerance Statistic is low, the predictor overlaps strongly with other predictors.

In simple words, the Tolerance Statistic answers this question: “After the other predictors explain this predictor, how much unique variance is still left?” A tolerance value close to 1 means the predictor is mostly independent of the other predictors. A tolerance value near 0 means the predictor is mostly redundant with other predictors. Low tolerance can inflate standard errors, weaken statistical power, and make regression coefficients unstable.

This guide uses a regression model predicting G3 final grade. The predictors include previous grades, study time, failures, absences, age, parental education variables, travel time, family relationship, free time, going out, and health. Because G1 and G2 are both previous grade variables, they naturally overlap. The Tolerance Statistic output confirms this: G1 and G2 have the lowest tolerance values, but their values remain above the .20 warning threshold.

Tolerance Statistic Formula and VIF Relationship

The Tolerance Statistic for a predictor is based on an auxiliary regression. In that auxiliary regression, one predictor is predicted from all the other predictors. The R² from that auxiliary regression is then used to calculate tolerance.

Here, R²_j is the R-squared value from predicting predictor X_j using all other predictors. If the other predictors explain a large part of X_j, then R²_j is high and the Tolerance Statistic becomes low.

The Tolerance Statistic is also the inverse of the Variance Inflation Factor:

Therefore, low tolerance and high VIF describe the same multicollinearity problem from opposite directions. If tolerance is low, VIF is high. If tolerance is high, VIF is low.

Null and Alternative Hypothesis for Tolerance Statistic

The Tolerance Statistic is a diagnostic rule rather than a traditional p-value hypothesis test. However, a hypothesis-style decision is useful for assumption-check reporting because it makes the result clear and consistent with other statistical guide posts.

Dataset and Regression Variables Used

The worked example uses a student performance regression model. The dependent variable is G3 final grade. The predictors are G1, G2, studytime, failures, absences, age, Medu, Fedu, traveltime, famrel, freetime, goout, and health. The Tolerance Statistic is calculated only for predictors, not for the dependent variable.

For broader descriptive review before regression, use descriptive statistics, frequency distribution, five-number summary, histogram interpretation, and box plot interpretation.

Verified SPSS Output Interpretation

The SPSS output provides the main evidence for the Tolerance Statistic decision. The regression model used G3 as the dependent variable and entered all 13 predictors. SPSS reports the Tolerance Statistic in the Coefficients table under Collinearity Statistics. The same table also reports VIF, which is the inverse of tolerance.

SPSS Model Summary and ANOVA

SPSS Tolerance Statistic and VIF Cross-check

SPSS Tolerance Statistic Decision Summary

SPSS Collinearity Diagnostics and Condition Index

SPSS also reports a collinearity diagnostics table. The table helps identify deeper collinearity structure beyond the basic Tolerance Statistic column. In this model, high condition index values appear, and one high-index dimension involves large variance proportions for G1 and G2. This supports the same practical conclusion: the main overlap is between the two previous grade predictors. The tolerance decision remains acceptable, but G1 and G2 should not be described as fully independent predictors.

Python Chart-by-Chart Interpretation

The Python charts show the Tolerance Statistic pattern visually. The main message is consistent across all figures: G1 and G2 have the lowest tolerance values, but all predictors remain above the common .20 threshold. Therefore, the model has acceptable multicollinearity under the selected tolerance decision rule.

Python Chart 1: Tolerance by Predictor

This chart ranks predictors by Tolerance Statistic. The lowest values are for G1 and G2, both around .238. These two predictors are the main variables to watch because they are previous grade measures and share substantial information. However, both values remain above the dashed .20 warning line. That means no predictor violates the common tolerance warning rule.

Most predictors have high tolerance values, especially health, famrel, absences, studytime and traveltime. High tolerance means these predictors are not strongly explained by the other predictors. The chart supports the final decision that the model has acceptable tolerance values, while still identifying G1 and G2 as the main caution variables.

Python Chart 2: VIF Cross-check

The VIF cross-check verifies the tolerance result from the opposite direction. Since VIF is the inverse of tolerance, the predictors with the lowest tolerance have the highest VIF. G2 and G1 have the highest VIF values, around 4.20, but they remain below the common VIF = 5 reference line. This confirms that the low tolerance values are elevated but still acceptable.

The VIF cross-check is important because many readers are more familiar with VIF than tolerance. In this model, both diagnostics tell the same story: no predictor crosses the warning threshold, but G1 and G2 have the most predictor overlap.

Python Chart 3: Predictor Correlation Heatmap

The predictor correlation heatmap explains why some predictors have lower tolerance. The strongest off-diagonal relationship is between G1 and G2. This relationship explains why both grade predictors have the lowest tolerance and highest VIF cross-check values. Another visible relationship is between Medu and Fedu, which explains their moderate tolerance values.

The heatmap is useful because the Tolerance Statistic is not only a number to report. It should be interpreted with predictor relationships. When predictors are strongly correlated, they share explanatory information. That shared information reduces tolerance and can make coefficient interpretation less stable.

Python Chart 4: Top Pairwise Predictor Correlations

This chart isolates the strongest pairwise predictor correlations. The strongest relationship is G1 vs G2, which is high enough to explain why these two variables have the lowest tolerance. The next strongest relationship is Medu vs Fedu. Other pairs are weaker and do not create serious tolerance problems.

This chart should not replace the tolerance table because tolerance can reflect combined overlap with several predictors. However, in this model the strongest pairwise relationship clearly explains the main tolerance pattern. The chart supports the conclusion that G1 and G2 require interpretation caution.

Python Chart 5: Tolerance vs VIF

This scatter plot directly shows that Tolerance Statistic and VIF are inverse diagnostics. Predictors with low tolerance appear with high VIF. G1 and G2 sit in the lower-tolerance, higher-VIF region. The remaining predictors cluster near high tolerance and low VIF.

This chart is a useful teaching figure because it shows why tolerance and VIF should not be treated as unrelated measures. They are two forms of the same collinearity information. A clean report may include both, but the interpretation should be consistent.

Python Chart 6: Regression Coefficients

This chart shows the regression coefficients after all predictors are entered together. G2 has the strongest positive coefficient, while failures has a negative coefficient. The coefficient chart explains why tolerance matters: if predictors overlap strongly, coefficient signs and standard errors can become unstable.

In this model, the Tolerance Statistic is acceptable, so the coefficients can be reported. However, G1 and G2 should still be interpreted as adjusted grade predictors. Their coefficients represent their contribution after the other grade variable and all other predictors are controlled.

Python Chart 7: Tolerance Decision Summary

The decision summary chart gives the final tolerance result. All 13 predictors fall in the acceptable tolerance category. No predictor falls below .20 and no predictor falls below .10. This chart is the easiest visual for a final report because it clearly shows that the regression model does not violate the selected tolerance threshold.

R Chart-by-Chart Validation

The R charts validate the same Tolerance Statistic analysis using a separate software workflow. The R charts match the Python and SPSS interpretation: G1 and G2 have the lowest tolerance values, all predictors remain above .20, and the final decision is acceptable multicollinearity under the selected tolerance rule.

R Chart 1: Tolerance by Predictor

The R tolerance chart reproduces the same ranking as Python and SPSS. G1 and G2 have the lowest values, while all other predictors are comfortably above the threshold. This validates the conclusion that the predictor set is acceptable but that grade variables require careful interpretation.

R Chart 2: VIF Cross-check

The R VIF cross-check confirms that no predictor exceeds the common VIF = 5 threshold. G1 and G2 are the highest VIF predictors because they are the lowest tolerance predictors. This confirms that the tolerance result is not a software artifact; it is the same inverse relationship shown in Python and SPSS.

R Chart 3: Predictor Correlation Heatmap

The R heatmap confirms the same predictor relationship structure. The strongest relationship is G1 with G2. Medu and Fedu also show a noticeable relationship. These correlations help explain why tolerance is lower for some variables than others.

R Chart 4: Top Pairwise Predictor Correlations

The R top-correlation chart again identifies G1 vs G2 as the strongest predictor pair. This supports the low tolerance values for G1 and G2. Since all tolerance values remain above .20, the relationship is important for interpretation but not severe enough to reject the model based on tolerance.

R Chart 5: Tolerance vs VIF

The R scatter plot confirms that tolerance and VIF move in opposite directions. G1 and G2 appear as low-tolerance and high-VIF predictors, while most other predictors appear as high-tolerance and low-VIF predictors. This validates the Python scatter plot and strengthens the interpretation.

R Chart 6: Regression Coefficients

The R coefficient chart confirms the same coefficient direction as the SPSS and Python results. G2 is the strongest positive predictor, G1 is also positive, and failures is negative. Since tolerance is acceptable, these coefficients can be reported, but the grade predictors should still be interpreted with caution.

R Chart 7: Tolerance Decision Summary

The R decision summary confirms that all predictors fall in the acceptable category. No predictor is below .20 and no predictor is below .10. This independent validation strengthens the final conclusion that the model has acceptable tolerance values.

SPSS, R, Python and Excel Workflows for Tolerance Statistic

The same Tolerance Statistic logic can be applied in SPSS, R, Python and Excel. SPSS reports tolerance directly in the regression Coefficients table. R and Python calculate tolerance from VIF or from auxiliary regression R² values. Excel can reproduce the same method manually by regressing each predictor on all other predictors.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Tolerance Statistic

SPSS Syntax for Tolerance Statistic

* Tolerance Statistic and VIF in SPSS.
* Dependent variable: G3.
* Predictors: G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health.

TITLE "Tolerance Statistic and VIF through SPSS collinearity diagnostics".

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health
  /STATISTICS COEFF OUTS R ANOVA COLLIN TOL
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN.

CORRELATIONS
  /VARIABLES=G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE.

DESCRIPTIVES VARIABLES=G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health
  /STATISTICS=MEAN STDDEV MIN MAX.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Tolerance-Statistic-SPSS-Output.pdf".

Python Code for Tolerance Statistic

import pandas as pd
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor

df = pd.read_csv("dataset.csv")

dependent = "G3"
predictors = [
    "G1", "G2", "studytime", "failures", "absences", "age",
    "Medu", "Fedu", "traveltime", "famrel", "freetime", "goout", "health"
]

model_data = df[[dependent] + predictors].apply(pd.to_numeric, errors="coerce").dropna()

X = model_data[predictors]
X_const = sm.add_constant(X)
y = model_data[dependent]

model = sm.OLS(y, X_const).fit()
print(model.summary())

rows = []
for i, variable in enumerate(X_const.columns):
    if variable == "const":
        continue

    vif_value = variance_inflation_factor(X_const.values, i)
    tolerance_value = 1 / vif_value

    rows.append({
        "predictor": variable,
        "Tolerance Statistic": tolerance_value,
        "VIF": vif_value,
        "Decision": "Acceptable" if tolerance_value >= 0.20 else "Review"
    })

tolerance_table = pd.DataFrame(rows).sort_values("Tolerance Statistic")
print(tolerance_table)

corr_matrix = X.corr()
print(corr_matrix)

R Code for Tolerance Statistic

# Tolerance Statistic in R

df <- read.csv("dataset.csv")

predictors <- c(
  "G1", "G2", "studytime", "failures", "absences", "age",
  "Medu", "Fedu", "traveltime", "famrel", "freetime", "goout", "health"
)

vars_needed <- c("G3", predictors)
df_model <- df[vars_needed]
df_model[] <- lapply(df_model, as.numeric)
df_model <- na.omit(df_model)

model <- lm(
  G3 ~ G1 + G2 + studytime + failures + absences + age +
    Medu + Fedu + traveltime + famrel + freetime + goout + health,
  data = df_model
)

summary(model)

library(car)
vif_values <- vif(model)

tolerance_table <- data.frame(
  predictor = names(vif_values),
  Tolerance_Statistic = 1 / as.numeric(vif_values),
  VIF = as.numeric(vif_values)
)

tolerance_table <- tolerance_table[order(tolerance_table$Tolerance_Statistic), ]
print(tolerance_table)

corr_matrix <- cor(df_model[predictors], use = "complete.obs")
print(corr_matrix)

Excel Formulas for Tolerance Statistic

Step 1:
Choose one predictor as the dependent variable in an auxiliary regression.

Example:
Predict G2 from G1, studytime, failures, absences, age, Medu, Fedu, traveltime, famrel, freetime, goout, health.

Step 2:
Get the R Square value from that auxiliary regression.

Step 3:
Calculate Tolerance Statistic:
=1-R_Square

Step 4:
Calculate VIF cross-check:
=1/(1-R_Square)

Step 5:
Interpret:
If tolerance is greater than or equal to .20, commonly acceptable.
If tolerance is below .20, review multicollinearity.
If tolerance is below .10, serious warning.

Example:
If auxiliary R Square = 0.762

Tolerance Statistic:
=1-0.762
=0.238

VIF:
=1/0.238
=4.202

APA Reporting Wording for Tolerance Statistic

When reporting Tolerance Statistic, include the threshold rule, the lowest tolerance value, the VIF cross-check, and the final decision. If the model contains related predictors such as G1 and G2, mention that their coefficients should be interpreted cautiously even when tolerance is acceptable.

APA-Style Tolerance Statistic Report

Full Model Report Sentence

Common Mistakes in Tolerance Statistic Interpretation

When to Use Tolerance Statistic

Use Tolerance Statistic whenever a regression model has multiple predictors and you need to check whether predictors overlap too strongly. It is especially important when the purpose is coefficient interpretation rather than only prediction.

Do not use Tolerance Statistic as a substitute for descriptive spread or practical importance. If you need variability comparison, use the coefficient of variation. If you need uncertainty intervals, use the confidence interval guide. If you need practical magnitude, use effect size.

Downloads and Resources for Tolerance Statistic

The SPSS output PDF below verifies the regression model, Tolerance Statistic values, VIF cross-check, collinearity diagnostics, predictor correlation matrix and descriptive statistics used in this guide.

Related Statistical Guides

Use these related guides to connect the Tolerance Statistic with regression assumptions, normality checks, variance checks, model diagnostics and reporting.